Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions

TL;DR

This paper proves the local existence and uniqueness of strong solutions for stochastic third grade fluid equations with Navier boundary conditions in bounded domains, using advanced stochastic analysis techniques.

Contribution

It introduces a novel approach combining cut-off approximation, stochastic compactness, and Yamada-Watanabe theorem for these complex fluid equations.

Findings

Existence of local strong solutions in Sobolev space H^3

Uniqueness of solutions up to a positive stopping time

Method applicable to non-Newtonian fluids with stochastic forcing

Abstract

This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness of the local (in time) solution, which corresponds to an addapted stochastic process with sample paths defined up to a certain positive stopping time, with values in the Sobolev space H^3. Our approach combines a cut-off approximation scheme, a stochastic compactness arguments and a general version of Yamada-Watanabe theorem. This leads to the existence of a local strong pathwise solution.